International Journal of Advanced Science and Technology
Vol. x, No. x, xxxxx, 2009
Linear Quadratic State Feedback Design for Switched Linear
Systems with Polytopic Uncertainties
Vu Trieu Minh and Ahmad Majdi Abdul Rani
Mechanical Engineering Department,
Universiti Teknologi PETRONAS (UTP),
Bandar Seri Iskandar, 31750 Tronoh, Perak Darul Ridzuan, Malaysia
[email protected] , [email protected]
Abstract
In this paper, we consider the design of stable linear switched systems for polytopic
uncertainties via their closed loop linear quadratic state feedback regulator. The closed loop
switched systems can stabilize unstable open loop systems or stable open loop systems but in
which there is no solution for a common Lyapunov matrix. For continuous time switched
linear systems, we show that if there exists solution in an associated Riccati equation for the
closed loop systems sharing one common Lyapunov matrix, the switched linear systems are
stable. For the discrete time switched systems, we derive an LMI to calculate a common
Lyapunov matrix and solution for the stable closed loop feedback systems. These closed loop
linear quadratic state feedback regulators guarantee the global asymptotical stability for any
switched linear systems with any switching signal sequence.
Keywords: Continuous time linear switched system, discrete time switched linear systems, linear quadratic
state feedback regulator, common Lyapunov matrix.
1. Introduction
A switched linear systems is hybrid dynamical system which consists of several
linear subsystems and a switching rule that decides which of switching rule is active at
each moment. In the last two decades, there has been increasing interest in stability
analysis and control design for switched systems in [1], [2], [3], [4], [5], [6], [7], [8],
[9] and [12] The motivation for studying switched systems is from the fact that many
practical systems are inherently multimodal. Many researchers have studied the use of
multiple models in adaptive control of both linear and nonlinear in which controllers
are switched depending on which model provides the least identification errors.
Stability results for such continuous time, switching control systems have been shown
for the linear [10] and for a certain nonlinear case [11]. The linear multiple model
switching is similar to the control of Markovian jump linear systems or the system has
models whose parameters change with respect to an underlying Markov chain.
This paper assumes that the switching signal can be designed by control engineers.
Hence, we investigate the design of stable linear switched systems for polytopic
uncertainties via their closed loop linear quadratic state feedback regulators. The closed
loop linear switched systems can stabilize unstable systems or stable systems but there
is no solution for a direct common Lyapunov matrix. It is clearly that the quadratic
stability requires for uncertain systems a quadratic Lyapunov function which guarantees
asymptotical stability for all uncertainties.
41
International Journal of Advanced Science and Technology
Vol. x, No. x, xxxxx, 2009
The outline of this paper is as follows. In section 2, we consider the linear quadratic
state feedback design for continuous time case. Section 3 presents the linear quadratic
state feedback design for discrete time case. Three examples are provided in section 2
and section 3 to illustrate the main ideas in each section. Finally in section 4,
conclusions are drawn and some directions of future research are discussed.
2. Linear Quadratic State Feedback Design for Continuous-Time Case
In this section, we consider the continuous-time switched linear system
x (t )  A ( x ,t ) x(t )
(2.1)
where x(t )   n is the state vector,  ( x, t ) is a switching rule defined by
 ( x, t ) :  n     {1, 2,..., N i } , and   denotes nonnegative real numbers. Therefore,
the switched system is composed of continuous time combination of
CSi : x (t )  Ai x(t ) for i  {1, 2,..., N i }
Here, we assume that CSi are uncertain polytopic type described as:
(2.2)
Nj
Ai   ij Aj for j  {1, 2,..., N j }
(2.3)
j 1
where N j are the number of the extreme points of the polytope (constant matrices)
Aj  { A1 , A2 ,..., AN j } and the weighting factors i  {i1 , i 2 ,..., iN j } belongs to
Nj
i :  ij  1 , ij  0
(2.4)
j 1
As indicated in [1], even if each of matrices Aj and all switched systems CSi are
globally stable with their eigenvalues being absolutely negative, there can exist a switching
sequence that destabilizes the close-loop dynamics. For all given stable matrices Aj , the
stability of the switched systems CSi is guaranteed if we can find out a common Lyapunove
matrix P .
Lemma 2.1: The switched linear systems CSi for stable polytopic uncertainties Aj can
guarantee the global asymptotical stability for any switched linear systems with any switching
signal sequence if there exists a common positive symmetric definite matrix P  P '  0 and
positive symmetric definite matrices Q j  Q 'j  0 such that Aj P  P A'j  Q j , j .
Proof: Since we assume that all matrices Aj are stable and the state update equations for
Nj
the linear switched systems (2.3) are x  Ai x  ( ij Aj ) x . For a positive Lyapunov function
j 1
Vi ( x)  x (t ) Px(t ) , we have always a negative time derivative Vi ( x )  0 , and the system is
stable for any switched systems with any switching signal sequence:
'
'
Nj
 Nj

 Nj
 Nj
Vi ( x)    ij Aj x  Px  x ' P   ij Aj x    ij x ' ( Aj P  PA'j ) x   ij x ' (Q j ) x  0
j 1
 j 1

 j 1
 j 1
42
(2.5)
International Journal of Advanced Science and Technology
Vol. x, No. x, xxxxx, 2009
The existence of a direct common Lyapunov matrix Aj P  PA'j  Q j among stable
matrices Aj and positive symmetric definite matrices Q j  Q 'j  0 can be searched with
quadratic stability of polytopic systems or directly solved with LMIs.
Example 2.1: Consider the switched linear systems CSi composed of four extreme points
 2 1 
 2 1
 0.2 0.5
 1 1
, A12  
, A13  
, and A14  
A11  
(2.6)




 1 1
 1 2 
 1 2 
 0.3 0.1 
All above matrices are stable (their eigenvaluse pi of those matrices are absolutely
negative):
 1
p11  0.05  0.3571i , p12  1  1i , p13    , and p14  2  1i
(2.7)
 3 
Searching with quadratic stability of polytopic systems, we find out a common Lyapunov
 0.8928 0.4107 
matrix for all four matrices P1a  
.
 0.4107 1.5454 
Solving directly with LMIs, we find out another similar common Lyapunov matrix
0.71311 0.2920 
P1b  
 . Here, we can conclude that the switched linear systems CSi are
 0.2920 1.0851 
stable with any switched linear systems and with any switching signal sequence.
It is difficult to find out a direct common Lyapunov matrix P in (2.5) for all extreme
points of the polytope Aj . In this example, if we only change A14 in equation (2.6) by a new a
 0.25 0.5
New
stable matrix A14New  
 with small eigenvalues p14  0.075  0.685i , there is no

1
0.1


solution for a common Lyapunov matrix for four stable matrices.
The assumption that all matrices Aj are stable seems usually unrealistic. Thus, how can
we deal with unstable systems or stable systems but in which there is no solution for a
common Lyapunov matrix?
For stabilizing all possible switched linear systems with any switching sequence, we
investigate the design of a close loop quadratic state feedback regulator, one of the most
common useful algorithms in control theory. The algorithm allows to change from unstable
open loop dynamics x (t )  Ax(t ) into some stable closed loop dynamics
x  ( A  BK ) x  AC x .
In the design of a linear quadratic state feedback regulator for a continuous time system,
we seek to find out the stable linear state feedback gain K in u   Kx for the state space
model x  Ax  Bu , where B is the controller input (matrix). The linear quadratic regulator
returns the solution P  P '  0 of the associated Riccati equation:
A' P  PA  ( PB) R 1 ( B ' P)  Q  0
(2.8)
where R  R '  0 and Q  Q '  0 are weighted matrices for input and state. The regulator

minimizes the quadratic cost function J (u )   ( x 'Qx  u ' Ru )dt , then the optimal linear
0
feedback gain K :
K  R 1 ( B ' P)
(2.9)
43
International Journal of Advanced Science and Technology
Vol. x, No. x, xxxxx, 2009
By fixing R and selecting upper hand a common Lyapunov matrix P , we can design
new stable closed loop matrices ACj  ( Aj  B j K j ) which stabilize all switched systems CSi
applied to closed loop matrices ACj with any switching signal sequence.
For simplicity, we set R  I , and the closed-loop common Lyapunov P  I , the
equations (2.8) and (2.9) become:
A'j  Aj  B j B 'j  Q j  0
(2.10)
and
K j  B 'j
(2.11)
Then, the close loop matrices are
ACj  Aj  B j B 'j
(2.12)
Lemma 2.2: The switched linear systems CSi for stable closed loop matrices ACj can
guarantee the global asymptotical stability for any switched linear systems with any switching
signal sequence if there exists solution for the control matrices B j  0 and the positive
symmetric matrices Q j  Q 'j  0 in the associated Riccati equation (2.10).
Proof: As indicated in Lemma 2.1, the switched systems CSi for stable closed loop
matrices ACj can guarantee the global asymptotical stability if there exists a positive
symmetric definite matrix (common Lyapunov matrix) P  P '  0 and positive symmetric
definite matrices Q j  Q 'j  0 such that ACj P  PACj '  Q j , j . Since we set the common
Lyapunov matrix P  I for all stable closed loop matrices ACj , the negative time derivative
Vi ( x)  0 then becomes ACj  ACj '  0 . From (2.12), we have:
ACj '  ACj  Aj  B j B 'j  A'j  B j B 'j  ( Aj  A'j )  2 B j B 'j
(2.13)
Replace A  Aj  B j B  Q j from equation (2.10) to equation (2.13) , we have:
'
j
'
j
ACj '  ACj  Q j  B j B 'j  0
(2.14)
So that for any positive Lyapunov function Vi ( x)  x ' (t ) Px(t ) of the stable closed loop
matrices AC , we have always a negative time derivative V ( x )  0 in (2.14), and the system is
j
i
stable with any switched linear systems and with any switching signal sequence CSi .
Example 2.2: This example is taken in [2]. Consider a switched linear system CSi
composed of four unstable matrices:
 1 2 
 1 1 
 1 2 
 1 1
, A22  
, A23  
, and A24  
A21  
(2.15)




 2 1
 1 1
 2 1
 1 1
Solving the Reccati equations in (2.8) with R  R '  I , and a common Lyapunov matrix
P  P '  I , we find out the solution for the corresponding control matrices B j  0 in (2.10)
as:
 0.9557 0.4551
 0.4804 0.0204 
, B22  
B21  

,
 0.4551 0.9557 
 0.0204 0.4804 
 0.9557 0.4551
 0.4804 0.0204 
B23  
, and B24  


 0.4551 0.9557 
 0.0204 0.4804 
44
(2.14)
International Journal of Advanced Science and Technology
Vol. x, No. x, xxxxx, 2009
and the corresponding positive symmetric matrices Q j  Q 'j  0 ,
 3.1349 3.1232 
 2.2354 2.0231
, Q22  
Q21  

,
 3.1232 3.1349 
 2.0231 2.2354 
 0.9557 0.4551
3.1349 2.0231
, and Q24  
Q23  


 0.4551 0.9557 
 2.0231 2.2354 
(2.14)
And then, the lemma 2.2 is hold, the switched linear systems CSi applied to stable
closed loop matrices ACj are stable with any switched linear systems and with any switching
signal sequence.
For stable open loop matrices in which we cannot find out a common Lyapunov matrix
as shown in example 2.1 with Aj  { A11 , A12 , A13 , A14New } , we can also re-design new stable
closed loop matrices that assure stability for any switched linear systems with any switching
signal sequence of CSi :
 0.4583 0.4859 
 1.2513 1.000 
A11C  
, A12C  

,
 0.3141 0.2007 
 1.000 1.2513
 2.2509 0.9996 
 0.5503 0.5912 
, A14C  
A13C  


 0.9996 2.2509 
 0.9088 0.3208 
(2.15)
3. Linear Quadratic State Feedback Design for Discrete-Time Case
In this section, we consider the discrete time switched linear systems
x(k  1)  A ( x , k ) x(k )
where
x(k )  
n
(3.1)
is the state vector,  ( x, k ) is a switching rule defined by
 ( x, k ) :     {1, 2,..., N i } , and   denotes nonnegative integers. Therefore, the
n

switched system is composed of discrete time combination of:
CSi : x(k  1)  Ai x(k ) for i  {1, 2,..., N i }
Similarly, we assume that CSi are uncertain polytopic type described as
(3.2)
Nj
Ai   ij Aj for j  {1, 2,..., N j }
(3.3)
j 1
where N j are the number of the extreme points of the polytope (constant matrices)
Aj  { A1 , A2 ,..., AN j } and the weighting factors i  {i1 , i 2 ,..., iN j } belongs to
Nj
i :  ij  1 , ij  0
(3.4)
j 1
As indicated in [1], even if each of matrices Aj and all switched systems CSi are
globally stable with their eigenvalues 0  pii  1 , there can exist a switching sequence that
destabilizes the close-loop dynamics. For all given stable matrices Aj , the stability of the
switched systems CSi is guaranteed if we can find out a common Lyapunove matrix P .
Lemma 3.1: The switched linear systems CSi for stable polytopic uncertainties A j can
guarantee the global asymptotical stability for any switched linear systems with any switching
45
International Journal of Advanced Science and Technology
Vol. x, No. x, xxxxx, 2009
signal sequence if there exists a common positive symmetric definite matrix P  P '  0 and a
 P PA'j  


scalar   0 such that  Aj P P
0   0, j .
 
0  I 

Proof: Since we assume that all matrices Aj are stable and the state update equations for
Nj
the linear switched systems (3.3) are x(k  1)  Ai x(k )  ( ij Aj ) x( k ) . For the stable
j 1
discrete time systems, we always have the Lyapunov function decreasing Vi (k )  x ' (k ) Px( k )
and Vi (k  1)  Vi ( k )  0 , and the system is stable for any switched systems with any
switching signal sequence since they share a common Lyapunov matrix P  P '  0 :
'
 Nj
  Nj

Vi (k  1)  Vi (k )    ij Aj x  P   ij Aj x   xPx  0  Aj PA'j  P  0
(3.5)
 j 1
  j 1

By adding a scalar   0 in equation (3.5), we have P  Ai' PAi   I  0 , or
P  ( Ai' P) P 1 ( PAi )  ( ) I  1 ( )  0 . And using Schur complement, this equation is equivalent
to the LMI in lemma 3.1.
Hence, the common Lyapunov matrix in lemma 3.1 is the solution to the following LMI:
 P PA'j  


0   0, j .
min  , subject to  Aj P P
P  0,   0
 
0  I 

The assumption that all matrices Aj are stable (their eigenvalues 0  pii  1 ) seems
usually unrealistic. How can we deal with unstable discrete systems or stable discrete systems
but in which there is not a common Lyapunov matrix?.
For stabilizing all switched linear systems with any switching signal sequence, we
investigate the design of a closed loop quadratic feedback regulator for these unstable
matrices. The regulator computes the optimal input that minimizes the objective function in

quadratic form at each sampling time J ( k )    x( k  i )Qx(k  i)  u (k  i ) Ru ( k  i ) by a
i 0
linear feedback control law u (k  1)  Ku ( k ) for the closed loop stable state feedback
x(k  1)  Aj x(k )  B j u (k ) , in which Q and R are symmetric positive weighting matrices.
The regulator allows to change from unstable open loop update x( k  1)  A j x(k ) into some
stable closed loop update x(k  1)  ( Aj  B j K j ) x(k )  ACj x(k ) .
Lemma 3.2: The switched linear systems CSi for stable closed loop matrices ACj can
guarantee the global asymptotical stability for any switched linear systems with any switching
signal sequence if there exists solution for a common positive symmetric Lyapunov matrix
46
International Journal of Advanced Science and Technology
Vol. x, No. x, xxxxx, 2009
P  P '  PL1 , positive symmetric matrices Q  Q '  0 , R  R '  0 , and matrice K j  Y j1  0

R
Y j' ( AjY j  B j )' (Y j Q1/ 2 )' 


Yj
 PL
0
0 
 0 , j .
and B j  0 satisfying an LMI 
( AjY j  B j ) 0
PL
0 


1/ 2
0
0
I
 (Y j Q )

Proof: Suppose there exists a Lyapunov function V ( x) with V (0)  0 and
V ( x(k ))  x(k )' Px( k ) with Lyapunov matrix P  P '  0 . The system will be stable if the
Lyapunov function is decreasing, that is, V ( x(k  1))  V ( x( k ))  0 . Suppose
Vi (k  1)  Vi (k )  x(k  1)' Px(k  1)  x(k ) Px(k )   x(k )Qx(k )  u (k ) Ru ( k ) , k
Then, we have P  ( Aj  B j K j ) P ( Aj  B j K j )  Q  K RK j  0 . Set PL  P
'
'
j
(3.6)
1
and
1
j
K j  Y , the above equation can be transformed as
( R)  Y j' (  PL ) 1 Y j  ( AjY j  B j )' ( PL ) 1 ( AjY j  B j )  (Y j Q1/ 2 )' ( I ) 1 (Y j Q1/ 2 )  0
(3.7)
Using the Schur complement, equation 3.7 is equivalent to the LMI in lemma 3.2. The
system becomes stable for any switched linear systems CSi and with any switching signal
sequence since all stable closed loop state feedback ACj  ( Aj  B j K j ) share one common
Lyapunov matrix, the solution of the LMI in lemma 3.2.
Example 3.2: This example is taken in [12]. Consider the following set of uncertain
0.90 0.80 
0.20 
discrete models with model 1: x(k  1)  
x( k )  

 u ( k ) and model 2:
0.30 0.80 
0.20 
0.85 0.75
0.10 
x(k  1)  
x( k )  

 u ( k ) . Applied the linear quadratic feedback as
0.35 0.75 
 0.10 
1 0 
described in [12] with the weighted matrices as stated in lemma 3.2: Q  
 and R  1 .
0 1 
Solved directly with LMIs, we can find out a common Lyapunov mantrix for this set:
 3.3611 0.2859 
P
  0 . Then this closed loop hybrid system is global asymptotical stable
 0.2859 1.2163 
for any switched linear systems with any switching signal sequence.
4. Conclusions
In this paper, we have considered the replacement of unstable linear switched systems
for polytopic uncertainties via their closed loop linear quadratic state feedback regulator. For
both continuous time and discrete time linear switched systems, if there exists solution for the
closed loop state feedback, the switched linear systems always guarantee the global
asymptotical stability for any switched linear systems with any switching signal sequence.
There are several important issues which should be studied in the future work. First, we
set the closed loop common Lyapunov matrix P  I in equation (2.10) and find solution of
input matrices B j for the stable closed loop matrices ACj  Aj  B j B 'j . Then, problems are
still open for general case to find both variable matrices
P and R j . For discrete time
switched linear systems, the stabilizability condition (3.7) is directly derived from the
Lyapunov function (3.6). We can also solve the solution for the state feedback gain K and
47
International Journal of Advanced Science and Technology
Vol. x, No. x, xxxxx, 2009
the Lyapunov matrix P via the associated discrete time Riccati equation using linear
quadratic regulator design for discrete time systems. Feasible solution of the state feedback
design with sufficient conditions that guarantees the parameter dependent Lyapunov function
is also needed further studies.
References
[1] Branicky, M., “Multiple Lyapunov Functions and Other Analysis Tools for Switched and Hybrid System”.
IEEE Trans. Auto. Contr. , 1998, Vol. 43, No. 4, pp. 475-482.
[2] Zhai, Guisheng, Hai Lin, and Pasos Antsaklis. “Quadratic Stabilizability of Switched Linear Systems with
Polytopic Uncertaities”. Int. J. Control, 2003, Vol. 76, pp 747-753.
[3] Raymond A., Michael S, Stefan P., and Bengt L. “Perspectives and Results on the Stability and Stabilization of
Hybrid Systems”. Proc IEEE, 2000, Vol. 88, pp. 1069-1082.
[4] Liberson D. “Stabilizing a Linear System with Finite State Hybrid Output Feedback”. Proc of MED99, 1999,
pp. 176-183.
[5] Vesely V. “Static Output Feedback Robust Controller Design via LMI Approach” Journal of Electrical
Engineering, 2005, Vol. 56, No. 1-2.
[6] Akar, M., Paul, A. Michael, G., Mitra, U. “Conditions n the Stability of a Class of Second-Order Switched
Systems”. IEEE Tran on Aut Col., 2006, Vol 51(2), pp. 338-340.
[7] Berg, R., and Pogromsky, J. “Convergent Design of Switched Linear Systems”. Proc. IFAC Conf on Analysis
and Design of Hybrid Systems, Italy, 2006, Vol. 2, pp 122-128.
[8] Kim K, and Park Y. “Sliding Mode Design via Quadratic Performance Optimization with Pole-Clustering
Constraint” SIAM J. Cont. Optim., 2004, Vol 43(2), pp. 670-684.
[9] Jianbo, L. and Robert E. “Optimal Hybrid Control for Structures”. Computer-Aided Civil and Infrastructure
Engineering, 1998, Vol. 13, pp. 405-414.
[10] Narendra, K. and Balakrishnan, J. “Adaptive Control Using Multiple Models”. IEEE Trans on Automatic
Control, 1997, Vol 42(2), pp. 171-187.
[11] Narendra K, and Xiang, C. “Adaptive Control of Simple Nonlinear Systems using Multiple Models”. Proc. of
American Cont. Conf. , 2002, A.K, pp. 1779-1784.
[12] Vu Trieu Minh and Nitin Afzulpurkar. “Robust Model Predictive Control for Input Saturated and Soften State
Constraints”. Asian Journal of Control, 2005, Vol. 7, No. 3. pp. 323-329.
Authors
Vu Trieu Minh is a senior lecturer at the Department of Mechanical
Engineering of Universiti Teknologi PETRONAS (UTP), Malaysia. He
obtained his Ph.D of Mechatronics from Asian Institute of Technology
(AIT) with specialization in Model Predictive Control. He has previously
worked in Vietnam, Thailand, Germany and Malaysia. He has authored
over twenty research papers in the field of advanced process control. His
current research interests are Dynamical Systems, Model Based Control
Algorithms, Advanced Process Control. He is a member of IEEE.
Ahmad Majdi Bin Abdul Rani is a senior lecturer at the Department
of Mechanical Engineering of Universiti Teknologi PETRONAS
(UTP), Malaysia. Currently he is the head of the mechanical
department. He holds a BEng and a MSc of Industrial Engineering in
Northern Illinois University, USA, Ph.D of Manufacturing in
Loughborough University, UK. His interests are Rapid Prototyping
and Tooling; Advanced Manufacturing System; CAD/CAM, CNC
Programming; Robotics and Automation.
48